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\newcommand{\pd}{\partial}

\title{Proteinenabtrennung durch SMB-Prozess}
\author{Xue Zhang}

\begin{document}
%\maketitle
%\tableofcontents
%\begin{abstract}
%
%\end{abstract}

\section{S\"aulen-Model und GG-Modell}(Komponenten-Index: Salz(S), Protein BSA(1), Protein MYO(2))
\subsection{S\"aulen-Modell}
\subsubsection{ED-Modell}

\begin{align}%MB
\dfrac{\pd c}{\pd t} & = D_L \cdot \frac{\varepsilon_b}{\varepsilon_t} \cdot \dfrac{\pd^2 c}{\pd z^2} - u \cdot \frac{\varepsilon_b}{\varepsilon_t} \cdot \dfrac{\pd c}{\pd z} - \dfrac{1 - \varepsilon_t}{\varepsilon_t} \dfrac{\pd q}{\pd t} = RI - F \dfrac{\pd q}{\pd t},
\quad \textrm{mit} \quad u = \dfrac{4\dot V}{\pi \cdot d^2 \cdot \varepsilon_b}
\quad \textrm{und} \quad F = \dfrac{1 - \varepsilon_t}{\varepsilon_t}, \quad z:0...L \nonumber\\
\xrightarrow{x:=z/L} %dc/dt
\dfrac{\pd c}{\pd t} & = \dfrac{D_L}{L^2} \cdot \frac{\varepsilon_b}{\varepsilon_t} \cdot \dfrac{\pd^2 c}{\pd x^2} - \dfrac{u}{L} \cdot \frac{\varepsilon_b}{\varepsilon_t} \cdot \dfrac{\pd c}{\pd x} - \dfrac{1 - \varepsilon_t}{\varepsilon_t} \dfrac{\pd q}{\pd t} = RI - F \dfrac{\pd q}{\pd t}
\quad \textrm{und} \quad x=0...1 \label{eq:MB_x}\\
\textrm{bei x=0: } & u(c^0_0-c_f) = \dfrac{D_L}{L} \dfrac{\pd c}{\pd x} \mid_{x=0} \nonumber\\
\textrm{bei x=1: } & \dfrac{\pd c}{\pd x} \mid_{x=1} = 0 \nonumber
\end{align}

\subsubsection{Diskretisierung mit Orthogonale Kollokation der
partialen Ableitung in x-Richtung fuer alle Strecken n:0...NE-1}

\begin{align}
& \dfrac{\pd c}{\pd x} = \sum^{NP+2}_i \sum^{NP+2}_j AZ^n_{i,j} \cdot c^n_j \textrm{ und } \dfrac{\pd^2 c}{\pd x^2} = \sum^{NP+2}_i \sum^{NP+2}_j BZ^n_{i,j} \cdot c^n_j \nonumber\\
& \dfrac{\pd c}{\pd x} \mid_{x_j} = \sum^{NP+2}_j AZ^n_{i,j} \cdot c^n_j \textrm{ und } \dfrac{\pd^2 c}{\pd x^2} \mid_{x_j} = \sum^{NP+2}_i \sum^{NP+2}_j BZ^n_{i,j} \cdot c^n_j \nonumber\\
\nonumber\\
& \textrm{mit i, j=1...NP+2; NP: Anzahl der inneren Kollokationsstellen} \nonumber\\
\nonumber\\
& \textrm{und Kontinuitaetbedingungen: } \dfrac{\pd c^n}{\pd x} \mid_{Np+1} = \dfrac{\pd c^{n+1}}{\pd x} \mid_{0} \quad \textrm{ und } \quad c^n_{Np+1}=c^{n+1}_0  \nonumber\\
\rightarrow \nonumber
\end{align}
\newpage

\section{Einzelkomponent-Protein}
\subsection{ED-SD-Modell}

\begin{align}%qS,q1
q_S & = -0.5Q+0.5(Q^2+4S\cdot c^2_S)^{1/2} = f_S(c_S)
      \label{eq:SD-Einzel-qS}\\
q_1 & = K_1 \cdot c^{-z_1}_S \cdot c_1 = f_1(c_S,c_1)
      \label{eq:SD-Einzel-q1}
\end{align}

\begin{align}%dqS,dq1
\xrightarrow{(\ref{eq:SD-Einzel-qS})}%dqS/dt
\dfrac{\pd q_S}{\pd c_S} & = 2S\cdot c_S (Q^2+4S\cdot c^2_S)^{-1/2} = \lambda_S \nonumber\\
\Longrightarrow \dfrac{dq_S}{dt} & = \dfrac{\pd q_S}{\pd c_S}\dfrac{\pd c_S}{\pd t} = \lambda_S\dfrac{\pd c_S}{\pd t}
                                   = \dot{f}_S(c_S,\dfrac{dc_s}{dt})
                                   \label{eq:SD-Einzel-dqS}\\
\xrightarrow{(\ref{eq:SD-Einzel-q1})}%dq1/dt
\dfrac{\pd q_1}{\pd c_S} & = - z_1 \cdot K_1 \cdot c^{-z_1-1}_S = \lambda_{1S} \nonumber\\
\dfrac{\pd q_1}{\pd c_1} & = K_1\cdot c^{-z_1}_S = \lambda_{11} \nonumber\\
\Longrightarrow \dfrac{dq_1}{dt} & = \dfrac{\pd q_1}{\pd
c_S}\dfrac{\pd c_S}{\pd t} + \dfrac{\pd q_1}{\pd c_1}\dfrac{\pd
c_1}{\pd t} = \lambda_{1S}\dfrac{\pd c_S}{\pd t} +
\lambda_{11}\dfrac{\pd c_1}{\pd t} =
\dot{f}_1(c_S,c_1,\dfrac{dc_s}{dt},\dfrac{dc_1}{dt})
\label{eq:SD-Einzel-dq1}
\end{align}

\begin{align}%dcS,dc1
\xrightarrow{(\ref{eq:MB_x})+(\ref{eq:SD-Einzel-dqS})}%dcS/dt
\dfrac{\pd c_S}{\pd t} & = RI_S-F\cdot \lambda_S\dfrac{\pd c_S}{\pd t} = \dfrac{RI_S}{1+F\cdot \lambda_S} \label{eq:cS}\\
\xrightarrow{(\ref{eq:MB_x})+(\ref{eq:SD-Einzel-dq1})}%dc1/dt
\dfrac{\pd c_1}{\pd t} & = RI_1 - F(\lambda_{1S}\dfrac{\pd c_S}{\pd
t} + \lambda_{11}\dfrac{\pd c_1}{\pd t}) = \dfrac{RI_1-\dfrac{F\cdot
\lambda_{1S}\cdot RI_S}{1+F\cdot \lambda_S}}{1+F\cdot\lambda_{11}}
\label{eq:c1}
\end{align}
\newpage

\subsubsection{ED-SMA-Modell}
\begin{align}
q_S &= \dfrac{\Lambda \cdot c_S}{c_S + c_1 \cdot (\sigma_1 + \nu_1) \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}} = \dfrac{\Lambda \cdot c_S}{c_S + c_1 \cdot \xi_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}} = \dfrac{\Lambda \cdot c_S}{Nenner}
     \label{eq:SMA-Einzel-qS}\\
q_1 &= \dfrac{\Lambda \cdot c_1 \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}}{c_S + c_1 \cdot (\sigma_1 + \nu_1) \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}} = \dfrac{\Lambda \cdot c_1 \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}}{c_S + c_1 \cdot \xi_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}} = \dfrac{\Lambda \cdot c_1 \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}}{Nenner}
     \label{eq:SMA-Einzel-q1}\\
\textrm{mit } \xi_1 & = (\sigma_1 + \nu_1) \cdot K_1, \quad Nenner =
c_S + c_1 \cdot \xi_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 -
1} \nonumber
\end{align}

\begin{align}
\xrightarrow{(\ref{eq:SMA-Einzel-qS})} 0 & = q_S + \xi_1 \cdot c_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1} - \Lambda = q_S + a \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1} - \Lambda \nonumber\\
\textrm{mit } & a = \xi_1 \cdot c_1 \nonumber\\
\textrm{sei } & f(q_S) = q_S + a \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1} - \Lambda
                       \label{eq:SMA-Einzel-fqS}\\
\longrightarrow & \dfrac{\pd f(q_S)}{\pd q_S} = 1 + a \cdot \nu_1
\cdot q_S^{\nu_1 - 1} \cdot c_S^{-\nu_1} > 0 \textrm{ mit positiven
Parameter, und alle c und q} \geq 0 \label{eq:SMA-Einzel-dfqS}\\
\longrightarrow & f(q_S = 0) = - \Lambda < 0 \nonumber
\end{align}

\begin{align}
\xrightarrow{(\ref{eq:SMA-Einzel-qS})} %dqS/dt
0 &= q_S + \xi_1 \cdot c_1 \cdot q^{\nu_1}_S \cdot c^{-\nu_1}_S - \Lambda = f_S(c_1, c_S, q_S) \nonumber\\
\dfrac{\pd f_S}{\pd c_1} &= \xi_1 \cdot q^{\nu_1}_S \cdot c^{-\nu_1}_S = \lambda_{S1} \nonumber\\
\dfrac{\pd f_S}{\pd c_S} &= \xi_1 \cdot c_1 \cdot (-\nu_1) \cdot c^{-\nu_1-1}_S \cdot q^{\nu_1}_S = - \xi_1 \cdot c_1 \cdot \nu_1 \cdot q^{\nu_1}_S \cdot c^{-(\nu_1+1)}_S = \lambda_{SS} \nonumber\\
\dfrac{\pd f_S}{\pd q_S} &= 1 + \xi_1 \cdot c_1 \cdot \nu_1 \cdot q^{\nu_1-1}_S \cdot c^{-\nu_1}_S = \lambda_{Sq} \nonumber\\
\nonumber\\
\rightarrow \dfrac{df_S}{dt} &= 0 = \dfrac{\pd f_S}{\pd c_1} \dfrac{\pd c_1}{\pd t} + \dfrac{\pd f_S}{\pd c_S} \dfrac{\pd c_S}{\pd t} + \dfrac{\pd f_S}{\pd q_S} \dfrac{\pd q_S}{\pd t} = \lambda_{S1} \dfrac{\pd c_1}{\pd t} + \lambda_{SS} \dfrac{\pd c_S}{\pd t} + \lambda_{Sq} \dfrac{\pd q_S}{\pd t} \nonumber\\
\Rightarrow \dfrac{\pd q_S}{\pd t} & = -
\dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} -
\dfrac{\lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} =
\dot{f}_S \left( c_1, c_S, q_S, \dfrac{\pd c_1}{\pd t}, \dfrac{\pd
c_S}{\pd t} \right) \label{eq:SMA-Einzel-dqS}
\end{align}

\begin{align}%dq1/dt
\xrightarrow{(\ref{eq:SMA-Einzel-q1})}
0 &= q_1 + q_1 \cdot c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S - \Lambda \cdot c_1 \cdot K_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S = f_1 (c_1, c_S, q_1, q_S) \nonumber\\
\dfrac{\pd f_1}{\pd c_1} & = q_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S - \Lambda \cdot K_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S \nonumber\\
                         & = q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S \cdot (q_1 \cdot \xi_1 - \Lambda \cdot K_1) = \lambda_{11} \nonumber\\
\dfrac{\pd f_1}{\pd c_S} & = q_1 \cdot c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot ( - \nu_1) \cdot c^{ - \nu_1 - 1}_S - \Lambda \cdot c_1 \cdot K_1 \cdot q^{\nu_1 - 1}_S \cdot (-\nu_1) \cdot c^{-\nu_1-1}_S \nonumber\\
                         & = c_1 \cdot \nu_1 \cdot q^{\nu_1 - 1}_S \cdot c^{-(\nu_1+1)}_S \cdot (\Lambda \cdot  K_1  - q_1 \cdot \xi_1) = \lambda_{1S} \nonumber\\
\dfrac{\pd f_1}{\pd q_S} & = q_1 \cdot c_1 \cdot \xi_1 \cdot (\nu_1 - 1) \cdot q^{\nu_1 - 2}_S \cdot c^{ - \nu_1}_S - \Lambda \cdot c_1 \cdot K_1 \cdot (\nu_1 - 1) \cdot q^{\nu_1 - 2}_S \cdot c^{ - \nu_1}_S = \lambda_{1q} \nonumber\\
                         & = (\nu_1 - 1) \cdot c_1 \cdot q^{\nu_1 - 2}_S \cdot c^{ - \nu_1}_S \cdot (q_1 \cdot \xi_1 - \Lambda \cdot K_1) = \lambda_{1q} \nonumber\\
\dfrac{\pd f_1}{\pd q_1} & = 1 + c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S = \lambda_{q} \nonumber\\
\nonumber
\rightarrow \dfrac{\pd f_1}{\pd t} & = 0 = \dfrac{\pd f_1}{\pd c_1} \dfrac{\pd c_1}{\pd t} + \dfrac{\pd f_1}{\pd c_S} \dfrac{\pd c_S}{\pd t} + \dfrac{\pd f_1}{\pd q_S} \dfrac{\pd q_S}{\pd t} + \dfrac{\pd f_1}{\pd q_1} \dfrac{\pd q_1}{\pd t} \nonumber\\
                       & = \lambda_{11} \dfrac{\pd c_1}{\pd t} + \lambda_{1S} \dfrac{\pd c_S}{\pd t} + \lambda_{1q} \left( - \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} - \dfrac{\lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} \right) + \lambda_{q} \dfrac{\pd q_1}{\pd t} \nonumber\\
                       & = \dfrac{\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{1S} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} + \lambda_{q} \dfrac{\pd q_1}{\pd t} \nonumber\\
\Rightarrow \dfrac{\pd q_1}{\pd t} & = \dfrac{ \lambda_{1q} \cdot \lambda_{S1} - \lambda_{11} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_1}{\pd t} + \dfrac{ \lambda_{1q} \cdot \lambda_{SS} - \lambda_{1S} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_S}{\pd t}
                                     \label{eq:SMA-Einzel-dq1}\\
                                   & = \dot{f}_1 \left( c_1, c_S, q_S, \dfrac{\pd c_1}{\pd t}, \dfrac{\pd c_S}{\pd t} \right) \nonumber
\end{align}
\newpage

\begin{align}%dcS,dc1
\xrightarrow{(\ref{eq:MB_x}) + (\ref{eq:SMA-Einzel-dqS})}
\dfrac{\pd c_S}{\pd t} & = RI_S - F \dfrac{\pd q_S}{\pd t} \nonumber\\
                       & = RI_S + F \left( \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} \right) \nonumber\\
\xrightarrow{(\ref{eq:MB_x}) + (\ref{eq:SMA-Einzel-dq1})}
\dfrac{\pd c_1}{\pd t} & = RI_1 - F \dfrac{\pd q_1}{\pd t} \nonumber\\
                       & = RI_1 + F \left( \dfrac{\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{S1}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{1S} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{SS}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_S}{\pd t} \right) \nonumber\\
\nonumber\\
\Longrightarrow RI_S & = - F \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} + \left( 1 - F \dfrac{\lambda_{SS}}{\lambda_{Sq}} \right) \dfrac{\pd c_S}{\pd t}
                       \label{eq:SMA-Einzel-dcS}\\
RI_1 & = \left( 1 + F \dfrac{ \lambda_{1q} \cdot \lambda_{S1} - \lambda_{11} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \right) \dfrac{\pd c_1}{\pd t} + F \dfrac{\lambda_{1q} \cdot \lambda_{SS} - \lambda_{1S} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_S}{\pd t}
       \label{eq:SMA-Einzel-dc1}\\
\nonumber\\
\longrightarrow \overrightarrow{RI} & = \mathbf{G} \cdot \overrightarrow{\dfrac{\pd c}{\pd t}}
                                      \label{eq:ED-SMA-Einzel}\\
\textrm{mit } \mathbf{G} & = \left( \begin{array}{ccc}
1 - F \dfrac{\lambda_{SS}}{\lambda_{Sq}}& - F \dfrac{\lambda_{S1}}{\lambda_{Sq}} \\
F \dfrac{\lambda_{1q} \cdot \lambda_{SS} - \lambda_{1S} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} & 1 + F \dfrac{\lambda_{1q} \cdot \lambda_{S1} - \lambda_{11} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \\
\end{array}\right) \nonumber\\
\textrm{ und } \overrightarrow{\dfrac{\pd c}{\pd t}} & = \left(
\begin{array}{c}
\dfrac{\pd c_S}{\pd t}\\
\dfrac{\pd c_1}{\pd t}
\end{array}\right)
\textrm{ und } \overrightarrow{RI} = \left( \begin{array}{c}
RI_S\\
RI_1
\end{array} \right)
\nonumber\\
\rightarrow \overrightarrow{\dfrac{\pd c}{\pd t}} &= \mathbf{G}^{ - 1} \cdot \overrightarrow{RI} \nonumber\\
\textrm{sei } \mathbf{G} &= \left( \begin{array}{cc}
a&b\\
c&d
\end{array} \right) \nonumber\\
\rightarrow \mathbf{G}^{ - 1} &= \left( \begin{array}{ccc}
d & -b\\
-c & a
\end{array} \right) \cdot \frac{1}{a \cdot d-b \cdot c}\nonumber
\end{align}
\newpage

\subsection{Mehrkomponenten-Proteinen}
\subsubsection{ED-SD-Modell}
\begin{align}%qS,q1,q2
q_S & = -0.5Q+0.5(Q^2+4S\cdot c^2_S)^{1/2}
      = f_S(c_S)
      \label{eq:SD-Mehr-qS}\\
q_1 & = K_1 \cdot c^{-z_1}_S \cdot c_1
      = f_1(c_S,c_1)
      \label{eq:SD-Mehr-q1}\\
q_2 & = K_2 \cdot c^{-z_2}_S \cdot c_2
      = f_2(c_S,c_2)
      \label{eq:SD-Mehr-q2}
\end{align}

\begin{align}%dqS,dq1,dq2
\xrightarrow{(\ref{eq:SD-Mehr-qS})}%dqS/dt
\dfrac{\pd q_S}{\pd c_S} &=2S\cdot c_S (Q^2+4S\cdot c^2_S)^{-1/2}
                          =\lambda_S
                          \nonumber\\
\Longrightarrow\dfrac{dq_S}{dt} & = \dfrac{\pd q_S}{\pd
c_S}\dfrac{\pd c_S}{\pd t}
                                  = \lambda_S\dfrac{\pd c_S}{\pd t}
                                  = \dot{f}_S(c_S,\dfrac{dc_s}{dt})
                                  \label{eq:SD-Mehr-dqS}\\
\xrightarrow{(\ref{eq:SD-Mehr-q1})}%dq1/dt
\dfrac{\pd q_1}{\pd c_S} & = - z_1 \cdot K_1 \cdot c^{-z_1-1}_S
                           = \lambda_{1S}
                           \nonumber\\
\dfrac{\pd q_1}{\pd c_1} & = K_1\cdot c^{-z_1}_S
                           =\lambda_{11}
                           \nonumber\\
\Longrightarrow\dfrac{dq_1}{dt} & = \dfrac{\pd q_1}{\pd
c_S}\dfrac{\pd c_S}{\pd t}+\dfrac{\pd q_1}{\pd c_1}\dfrac{\pd
c_1}{\pd t}
                                  = \lambda_{1S}\dfrac{\pd c_S}{\pd t}+\lambda_{11}\dfrac{\pd c_1}{\pd t}
                                  = \dot{f}_1(c_S,c_1,\dfrac{dc_s}{dt},\dfrac{dc_1}{dt})
                                 \label{eq:SD-Mehr-dq1}\\
\xrightarrow{(\ref{eq:SD-Mehr-q2})}%dq2/dt
\dfrac{\pd q_2}{\pd c_S} & = - z_2 \cdot K_2 \cdot c^{-z_2-1}_S
                           =\lambda_{2S}
                           \nonumber\\
\dfrac{\pd q_2}{\pd c_2} & = K_2\cdot c^{-z_2}_S
                           = \lambda_{22}
                           \nonumber\\
\Longrightarrow \dfrac{dq_2}{dt} &= \dfrac{\pd q_2}{\pd
c_S}\dfrac{\pd c_S}{\pd t}+\dfrac{\pd q_2}{\pd c_2}\dfrac{\pd
c_2}{\pd t}
                                 =\lambda_{2S}\dfrac{\pd c_S}{\pd t}+\lambda_{22}\dfrac{\pd c_2}{\pd t}
                                 =\dot{f}_2(c_S,c_2,\dfrac{dc_s}{dt},\dfrac{dc_2}{dt})
                                 \label{eq:SD-Mehr-dq2}
\end{align}

\begin{align}%dcS,dc1,dc2
\xrightarrow{(\ref{eq:MB_x})+(\ref{eq:SD-Mehr-dqS})}%dcS/dt
\dfrac{\pd c_S}{\pd t} &=RI_S-F\cdot \lambda_S\dfrac{\pd c_S}{\pd t}
                        =\dfrac{RI_S}{1+F\cdot \lambda_S}
                        \label{eq:SMA-Mehr-cS}\\
\xrightarrow{(\ref{eq:MB_x})+(\ref{eq:SD-Mehr-dq1})}%dc1/dt
\dfrac{\pd c_1}{\pd t} &=RI_1-F(\lambda_{1S}\dfrac{\pd c_S}{\pd
t}+\lambda_{11}\dfrac{\pd c_1}{\pd t})
                        =\dfrac{RI_1-\dfrac{F\cdot \lambda_{1S}\cdot RI_S}{1+F\cdot \lambda_S}}{1+F\cdot\lambda_{11}}
                        \label{eq:SMA-Mehr-c1}\\
\xrightarrow{(\ref{eq:MB_x})+(\ref{eq:SD-Mehr-dq2})}%dc2/dt
\dfrac{\pd c_2}{\pd t} &=RI_2-F(\lambda_{2S}\dfrac{\pd c_S}{\pd
t}+\lambda_{22}\dfrac{\pd c_2}{\pd t})
                        =\dfrac{RI_2-\dfrac{F\cdot\lambda_{2S}\cdot RI_S}{1+F\cdot\lambda_S}}{1+F\cdot\lambda_{22}}
                        \label{eq:SMA-Mehr-c2}
\end{align}
\newpage

\subsubsection{ED-SMA-Modell}
\begin{align}%qS,q1,q2
q_S &= \dfrac{\Lambda \cdot c_S}{c_S + c_1 \cdot (\sigma_1 + \nu_1) \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + c_2 \cdot (\sigma_2 + \nu_2) \cdot K_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1}} \label{eq:SMA-Mehr-qS}\\
    &= \dfrac{\Lambda \cdot c_S}{c_S + c_1 \cdot \xi_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + c_2 \cdot \xi_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1}} \nonumber\\
    &= \dfrac{\Lambda \cdot c_S}{Nenner} \nonumber\\
q_1 &= \dfrac{\Lambda \cdot c_1 \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}}{c_S + c_1 \cdot (\sigma_1 + \nu_1) \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + c_2 \cdot (\sigma_2 + \nu_2) \cdot K_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1}} \label{eq:SMA-Mehr-q1}\\
    &= \dfrac{\Lambda \cdot c_1 \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}}{c_S + c_1 \cdot \xi_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + c_2 \cdot \xi_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1}} \nonumber\\
    &= \dfrac{\Lambda \cdot c_1 \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1}}{Nenner} \nonumber\\
q_2 &= \dfrac{\Lambda \cdot c_2 \cdot K_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1}}{c_S + c_1 \cdot (\sigma_1 + \nu_1) \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + c_2 \cdot (\sigma_2 + \nu_2) \cdot K_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1}} \label{eq:SMA-Mehr-q2}\\
    &= \dfrac{\Lambda \cdot c_2 \cdot K_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1}}{c_S + c_1 \cdot \xi_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + c_2 \cdot \xi_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1}} \nonumber\\
    &= \dfrac{\Lambda \cdot c_2 \cdot K_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1}}{Nenner} \nonumber\\
\nonumber\\
\textrm{mit} \quad \xi_1 &= (\sigma_1 + \nu_1) \cdot K_1,
\quad \xi_2 = (\sigma_2 + \nu_2) \cdot K_2 \nonumber\\ %\left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1} \nonumber\\
Nenner %&= c_S + c_1 (\sigma_1 + \nu_1) K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + c_2 (\sigma_2 + \nu_2) K_2 \cdot \left( \dfrac{q_S}{c_S} \right) ^{\nu_2 - 1} \nonumber\\
       &= c_S + c_1 \cdot \xi_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + c_2 \cdot \xi_2 \cdot \left( \dfrac{q_S}{c_S} \right) ^{\nu_2 - 1} \nonumber
\end{align}
\newpage

\begin{align}
\xrightarrow{(\ref{eq:SMA-Mehr-qS})}
0&= q_S + \xi_1 \cdot \dfrac{c_1}{c^{\nu_1}_S} \cdot q^{\nu_1}_S + \xi_2 \cdot \dfrac{c_2}{c^{\nu_2}_S} \cdot q^{\nu_2}_S - \Lambda \label{eq:q_Salz}\\
 &= q_S + a \cdot q_S^{\nu_1} + b \cdot q^{\nu_2}_S - \Lambda \nonumber\\
\textrm{mit } a &= \xi_1 \cdot \dfrac{c_1}{c^{\nu_1}_S}
\textrm{ und } b = \xi_2 \cdot \dfrac{c_2}{c^{\nu_2}_S} \nonumber\\
\textrm{sei } &f(q_S) = q_S + a \cdot q_S^{\nu_1} + b \cdot q^{\nu_2}_S - \Lambda \nonumber\\
\rightarrow & \dfrac{\pd f(q_S)}{\pd q_S} = 1 + a \cdot \nu_1 \cdot q_S^{\nu_1 - 1} + b \cdot \nu_2 \cdot q_S^{\nu_2-1} > 0 \nonumber\\
&\textrm{ mit positiven Parameter und alle c und q } \geq 0 \nonumber\\
&\textrm{ und } f(q_S = 0) = - \Lambda < 0 \nonumber
\end{align}

\begin{align}
\xrightarrow{(\ref{eq:SMA-Mehr-qS})} %SMA-Mehr-qSdqS/dt
0 &= q_S + \xi_1 \cdot c_1 \cdot c^{-\nu_1}_S \cdot q^{\nu_1}_S + \xi_2 \cdot c_2 \cdot c^{-\nu_2}_S \cdot q^{\nu_2}_S - \Lambda \nonumber\\
  &= f_S(c_1, c_2, c_S, q_S) \nonumber\\
\dfrac{\pd f_S}{\pd c_1} &= \xi_1 \cdot c^{-\nu_1}_S \cdot q^{\nu_1}_S \nonumber\\
                         &= \lambda_{S1} \nonumber\\
\dfrac{\pd f_S}{\pd c_2} &= \xi_2 \cdot c^{-\nu_2}_S \cdot q^{\nu_2}_S \nonumber\\
                         &= \lambda_{S2} \nonumber\\
\dfrac{\pd f_S}{\pd c_S} &= \xi_1 \cdot c_1 \cdot (-\nu_1) \cdot c^{-\nu_1-1}_S \cdot q^{\nu_1}_S + \xi_2 \cdot c_2 \cdot (-\nu_2) \cdot c^{-\nu_2-1}_S \cdot q^{\nu_2}_S \nonumber\\
                         &= - \xi_1 \cdot c_1 \cdot \nu_1 \cdot c^{-\nu_1-1}_S \cdot q^{\nu_1}_S - \xi_2 \cdot c_2 \cdot \nu_2 \cdot c^{-\nu_2-1}_S \cdot q^{\nu_2}_S \nonumber\\
                         &= \lambda_{SS} \nonumber\\
\dfrac{\pd f_S}{\pd q_S} &= 1 + \xi_1 \cdot c_1 \cdot c^{-\nu_1}_S \cdot \nu_1 \cdot q^{\nu_1-1}_S + \xi_2 \cdot c_2 \cdot c^{-\nu_2}_S \cdot \nu_2 \cdot q^{\nu_2-1}_S \nonumber\\
                         &= \lambda_{Sq} \nonumber\\
\nonumber\\
\rightarrow \dfrac{df_S}{dt} &= 0 = \dfrac{\pd f_S}{\pd c_1} \dfrac{\pd c_1}{\pd t} + \dfrac{\pd f_S}{\pd c_2} \dfrac{\pd c_2}{\pd t} + \dfrac{\pd f_S}{\pd c_S} \dfrac{\pd c_S}{\pd t} + \dfrac{\pd f_S}{\pd q_S} \dfrac{\pd q_S}{\pd t} \nonumber\\
                             &= \lambda_{S1} \dfrac{\pd c_1}{\pd t} + \lambda_{S2} \dfrac{\pd c_2}{\pd t} + \lambda_{SS} \dfrac{\pd c_S}{\pd t} + \lambda_{Sq} \dfrac{\pd q_S}{\pd t} \nonumber\\
\Rightarrow \dfrac{\pd q_S}{\pd t} &= - \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} - \dfrac{\lambda_{S2}}{\lambda_{Sq}} \dfrac{\pd c_2}{\pd t} - \dfrac{\lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} \label{eq:SMA-Mehr-dqS}\\
                                   &= \dot{f}_S \left( c_1, c_2, c_S, q_S, \dfrac{\pd c_1}{\pd t}, \dfrac{\pd c_2}{\pd t}, \dfrac{\pd c_S}{\pd t} \right) \nonumber
\end{align}
\newpage
\begin{align}%dq1/dt
\xrightarrow{(\ref{eq:SMA-Mehr-q1})}
0 &= q_1 \cdot c_S + q_1 \cdot c_1 \cdot \xi_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + q_1 \cdot c_2 \cdot \xi_2 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1} - \Lambda \cdot c_1 \cdot K_1 \cdot \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} \nonumber\\
0 &= q_1 + q_1 \cdot c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S + q_1 \cdot c_2 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot c^{ - \nu_2}_S - \Lambda \cdot c_1 \cdot K_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1} \nonumber\\
  &= f_1 (c_1, c_2, c_S, q_1, q_S) \nonumber\\
\dfrac{\pd f_1}{\pd c_1} &= q_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S - \Lambda \cdot K_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1} \nonumber\\
                         &= \lambda_{11} \nonumber\\
\dfrac{\pd f_1}{\pd c_2} &= q_1 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot c^{ - \nu_2}_S \nonumber\\
                         &= \lambda_{12} \nonumber\\
\dfrac{\pd f_1}{\pd c_S} &= q_1 \cdot c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot ( - \nu_1) \cdot c^{ - \nu_1 - 1}_S + q_1 \cdot c_2 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot ( - \nu_2) \cdot c^{ - \nu_2 - 1}_S - \Lambda \cdot c_1 \cdot K_1 \cdot q^{\nu_1 - 1}_S \cdot ( - \nu_1) \cdot c^{ - \nu_1} \nonumber\\
                         &= - q_1 \cdot c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot \nu_1 \cdot c^{ - \nu_1 - 1}_S - q_1 \cdot c_2 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot \nu_2 \cdot c^{ - \nu_2 - 1}_S + \Lambda \cdot c_1 \cdot K_1 \cdot q^{\nu_1 - 1}_S \cdot \nu_1 \cdot c^{ - \nu_1} \nonumber\\
                         &= \lambda_{1S} \nonumber\\
\dfrac{\pd f_1}{\pd q_S} &= q_1 \cdot c_1 \cdot \xi_1 \cdot (\nu_1 - 1) \cdot q^{\nu_1 - 2}_S \cdot c^{ - \nu_1}_S + q_1 \cdot c_2 \cdot \xi_2 \cdot (\nu_2 - 1) \cdot q^{\nu_2 - 2}_S \cdot c^{ - \nu_2}_S - \Lambda \cdot c_1 \cdot K_1 \cdot (\nu_1 - 1) \cdot q^{\nu_1 - 2}_S \cdot c^{ - \nu_1}_S \nonumber\\
                         &= \lambda_{1q} \nonumber\\
\dfrac{\pd f_1}{\pd q_1} &= 1 + c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S + c_2 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot c^{ - \nu_2}_S \nonumber\\
                         &= \lambda_{q} \nonumber\\
\rightarrow
\dfrac{\pd f_1}{\pd t} &= 0 = \dfrac{\pd f_1}{\pd c_1} \dfrac{\pd c_1}{\pd t} + \dfrac{\pd f_1}{\pd c_2} \dfrac{\pd c_2}{\pd t} + \dfrac{\pd f_1}{\pd c_S} \dfrac{\pd c_S}{\pd t} + \dfrac{\pd f_1}{\pd q_S} \dfrac{\pd q_S}{\pd t} + \dfrac{\pd f_1}{\pd q_1} \dfrac{\pd q_1}{\pd t} \nonumber\\
                       &= \lambda_{11} \dfrac{\pd c_1}{\pd t} + \lambda_{12} \dfrac{\pd c_2}{\pd t} + \lambda_{1S} \dfrac{\pd c_S}{\pd t} + \lambda_{1q} \left( - \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} - \dfrac{\lambda_{S2}}{\lambda_{Sq}} \dfrac{\pd c_2}{\pd t} - \dfrac{\lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} \right) + \lambda_{q} \dfrac{\pd q_1}{\pd t} \nonumber\\
                       &= \dfrac{\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} +  \dfrac{\lambda_{12} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{S2}}{\lambda_{Sq}} \dfrac{\pd c_2}{\pd t} + \dfrac{\lambda_{1S} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} + \lambda_{q} \dfrac{\pd q_1}{\pd t} \nonumber\\
\Rightarrow \dfrac{\pd q_1}{\pd t} &= \dfrac{ \lambda_{1q} \cdot \lambda_{S1} - \lambda_{11} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{1q} \cdot \lambda_{S2} - \lambda_{12} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_2}{\pd t} + \dfrac{ \lambda_{1q} \cdot \lambda_{SS} - \lambda_{1S} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_S}{\pd t} \label{eq:SMA-Mehr-dq1}\\
                                   &= \dot{f}_1 \left( c_1, c_2, c_S, q_S, \dfrac{\pd c_1}{\pd t}, \dfrac{\pd c_2}{\pd t}, \dfrac{\pd c_S}{\pd t} \right) \nonumber
\end{align}
\newpage
\begin{align}%dq2/dt
\xrightarrow{(\ref{eq:SMA-Mehr-q2})}
0 &= q_2 \cdot c_S + q_2 \cdot c_1 \cdot \xi_1 \left( \dfrac{q_S}{c_S} \right)^{\nu_1 - 1} + q_2 \cdot c_2 \cdot \xi_2 \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1} - \Lambda \cdot c_2 \cdot K_2 \left( \dfrac{q_S}{c_S} \right)^{\nu_2 - 1} \nonumber\\
0 &= q_2 + q_2 \cdot c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S + q_2 \cdot c_2 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot c^{ - \nu_2}_S - \Lambda \cdot c_2 \cdot K_2 \cdot q^{\nu_2 - 1}_S \cdot c^{ - \nu_2}_S \nonumber\\
  &= f_2 (c_1, c_2, c_S, q_2, q_S) \nonumber\\
\dfrac{\pd f_2}{\pd c_1} &= q_2 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S \nonumber\\
                         &= \lambda_{21} \nonumber\\
\dfrac{\pd f_2}{\pd c_2} &= q_2 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot c^{ - \nu_2}_S - \Lambda \cdot K_2 \cdot q^{\nu_2 - 1}_S \cdot c^{ - \nu_2}_S \nonumber\\
                         &= \lambda_{22} \nonumber\\
\dfrac{\pd f_2}{\pd c_S} &= q_2 \cdot c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot ( - \nu_1) \cdot c^{ - \nu_1 - 1}_S + q_2 \cdot c_2 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot ( - \nu_2) \cdot c^{ - \nu_2 - 1}_S - \Lambda \cdot c_2 \cdot K_2 \cdot q^{\nu_2 - 1}_S \cdot ( - \nu_2 )\cdot c^{ - \nu_2 - 1}_S \nonumber\\
                         &= - q_2 \cdot c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot \nu_1 \cdot c^{ - \nu_1 - 1}_S - q_2 \cdot c_2 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot  \nu_2 \cdot c^{ - \nu_2 - 1}_S + \Lambda \cdot c_2 \cdot K_2 \cdot q^{\nu_2 - 1}_S \cdot \nu_2 \cdot c^{ - \nu_2 - 1}_S \nonumber\\
                         &= \lambda_{2S} \nonumber\\
\dfrac{\pd f_2}{\pd q_S} &= q_2 \cdot c_1 \cdot \xi_1 \cdot (\nu_1 - 1) \cdot q^{\nu_1 - 2}_S \cdot c^{ - \nu_1}_S + q_2 \cdot c_2 \cdot \xi_2 \cdot (\nu_2 - 1) \cdot q^{\nu_2 - 2}_S \cdot c^{ - \nu_2}_S - \Lambda \cdot c_2 \cdot K_2 \cdot (\nu_2 - 1) \cdot q^{\nu_2 - 2}_S \cdot c^{ - \nu_2}_S \nonumber\\
                         &= \lambda_{2q} \nonumber\\
\dfrac{\pd f_2}{\pd q_2} &= 1 + c_1 \cdot \xi_1 \cdot q^{\nu_1 - 1}_S \cdot c^{ - \nu_1}_S + c_2 \cdot \xi_2 \cdot q^{\nu_2 - 1}_S \cdot c^{ - \nu_2}_S - \Lambda \cdot c_2 \cdot K_2 \cdot q^{\nu_2 - 1}_S \cdot c^{ - \nu_2}_S \nonumber\\
                         &= \lambda_{q} \nonumber\\
\rightarrow \dfrac{\pd f_2}{\pd t} &= 0 = \dfrac{\pd f_2}{\pd c_1} \dfrac{\pd c_1}{\pd t} + \dfrac{\pd f_2}{\pd c_2} \dfrac{\pd c_2}{\pd t} + \dfrac{\pd f_2}{\pd c_S} \dfrac{\pd c_S}{\pd t} + \dfrac{\pd f_2}{\pd q_S} \dfrac{\pd q_S}{\pd t} + \dfrac{\pd f_2}{\pd q_2} \dfrac{\pd q_2}{\pd t} \nonumber\\
                                   &= \lambda_{21} \dfrac{\pd c_1}{\pd t} + \lambda_{22} \dfrac{\pd c_2}{\pd t} + \lambda_{2S} \dfrac{\pd c_S}{\pd t} + \lambda_{2q} \left( - \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} - \dfrac{\lambda_{S2}}{\lambda_{Sq}} \dfrac{\pd c_2}{\pd t} - \dfrac{\lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} \right) + \lambda_{q} \dfrac{\pd q_2}{\pd t} \nonumber\\
                                   &= \dfrac{\lambda_{21} \cdot \lambda_{Sq} - \lambda_{2q} \cdot \lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} +  \dfrac{\lambda_{22} \cdot \lambda_{Sq} - \lambda_{2q} \cdot \lambda_{S2}}{\lambda_{Sq}} \dfrac{\pd c_2}{\pd t} + \dfrac{\lambda_{2S} \cdot \lambda_{Sq} - \lambda_{2q} \cdot \lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} + \lambda_{q} \dfrac{\pd q_2}{\pd t} \nonumber\\
\Rightarrow \dfrac{\pd q_2}{\pd t} &= \dfrac{\lambda_{2q} \cdot \lambda_{S1} - \lambda_{21} \cdot \lambda_{Sq} }{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{2q} \cdot \lambda_{S2} - \lambda_{22} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_2}{\pd t} + \dfrac{\lambda_{2q} \cdot \lambda_{SS} - \lambda_{2S} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_S}{\pd t} \label{eq:SMA-Mehr-dq2}\\
                                   &= \dot{f}_2 \left( c_1, c_2, c_S, q_S, \dfrac{\pd c_1}{\pd t}, \dfrac{\pd c_2}{\pd t}, \dfrac{\pd c_S}{\pd t} \right) \nonumber
\end{align}
\newpage
\subsection{ED-SMA-Modell}
\begin{align}%dcS,dc1,dc2
\xrightarrow{(\ref{eq:MB_x}) + (\ref{eq:SMA-Mehr-dqS})} \nonumber\\ %dcS/dt
\dfrac{\pd c_S}{\pd t} &= RI_S - F \dfrac{\pd q_S}{\pd t} \nonumber\\
                       &= RI_S + F \left( \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{S2}}{\lambda_{Sq}} \dfrac{\pd c_2}{\pd t} + \dfrac{\lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} \right) \nonumber\\%\label{eq:dc_S}\\
                       %&= \dfrac{RI_S}{1 - F \dfrac{\lambda_{SS}}{\lambda_{Sq}}} + \dfrac{F \dfrac{\lambda_{S1}}{\lambda_{Sq}}}{1 - F \dfrac{\lambda_{SS}}{\lambda_{Sq}}} \dfrac{\pd c_1}{\pd t} + \dfrac{F \dfrac{\lambda_{S2}}{\lambda_{Sq}}}{1 - F \dfrac{\lambda_{SS}}{\lambda_{Sq}}} \dfrac{\pd c_2}{\pd t} \nonumber\\
                       %&= \dfrac{RI_S \cdot \lambda_{Sq}}{\lambda_{Sq} - F \cdot \lambda_{SS}} + \dfrac{F \cdot \lambda_{S1}}{\lambda_{Sq} - F \cdot \lambda_{SS}} \dfrac{\pd c_1}{\pd t} + \dfrac{F \cdot \lambda_{S2}}{\lambda_{Sq} - F \cdot \lambda_{SS}} \dfrac{\pd c_2}{\pd t} \\
                       %&= RI_S - F \left( \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t}+\dfrac{\lambda_{S2}}{\lambda_{Sq}}\dfrac{\pd c_2}{\pd t}+\dfrac{\lambda_{SS}}{\lambda_{Sq}}\dfrac{\pd c_S}{\pd t} \right) \nonumber\\
                       %&= \dfrac{RI_S}{1 + F\dfrac{\lambda_{SS}}{\lambda_{Sq}}}-\dfrac{F\dfrac{\lambda_{S1}}{\lambda_{Sq}}}{1+F\dfrac{\lambda_{SS}}{\lambda_{Sq}}}\dfrac{\pd c_1}{\pd t}-\dfrac{F\dfrac{\lambda_{S2}}{\lambda_{Sq}}}{1+F\dfrac{\lambda_{SS}}{\lambda_{Sq}}}\dfrac{\pd c_2}{\pd t} \\
\xrightarrow{(\ref{eq:MB_x}) + (\ref{eq:SMA-Mehr-dq1})} \nonumber\\ %dc1/dt
\dfrac{\pd c_1}{\pd t} &= RI_1 - F \dfrac{\pd q_1}{\pd t} \nonumber\\
                       &= RI_1 + F \left( \dfrac{\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{S1}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{12} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{S2}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_2}{\pd t} + \dfrac{\lambda_{1S} \cdot \lambda_{Sq} - \lambda_{1q} \cdot \lambda_{SS}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_S}{\pd t} \right) \nonumber\\ %\label{eq:dc_1}\\
                       %&= \dfrac{RI_1}{\dfrac{\lambda_{Sq} (1 - \lambda_{1q}) - F (\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{S1})}{\lambda_{Sq} (1 - \lambda_{1q})}} + \dfrac{ \dfrac{F (\lambda_{12} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{S2})}{\lambda_{Sq} (1 - \lambda_{1q})}}{\dfrac{\lambda_{Sq} (1 - \lambda_{1q}) - F (\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{S1})}{\lambda_{Sq} (1 - \lambda_{1q})}} \dfrac{\pd c_2}{\pd t} + \dfrac{\dfrac{F (\lambda_{1S} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{SS})}{\lambda_{Sq} (1 - \lambda_{1q})}}{\dfrac{\lambda_{Sq} (1 - \lambda_{1q}) - F (\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{S1})}{\lambda_{Sq} (1 - \lambda_{1q})}} \dfrac{\pd c_S}{\pd t} \nonumber\\
                       %&= \dfrac{RI_1 \cdot \lambda_{Sq} (1 - \lambda_{1q})}{\lambda_{Sq} (1 - \lambda_{1q}) - F (\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{S1})} \nonumber\\
                       %&+ \dfrac{ F (\lambda_{12} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{S2})}{\lambda_{Sq} (1 - \lambda_{1q}) - F (\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{S1})} \dfrac{\pd c_2}{\pd t} \nonumber\\
                       %&+ \dfrac{F (\lambda_{1S} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{SS})}{\lambda_{Sq} (1 - \lambda_{1q}) - F (\lambda_{11} \cdot \lambda_{Sq} - \lambda_{1q_S} \cdot \lambda_{S1})} \dfrac{\pd c_S}{\pd t} \\
                       %&= RI_1 + F \left( \dfrac{\lambda_{11}}{\lambda_{1q}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{12}}{\lambda_{1q}} \dfrac{\pd c_2}{\pd t} + \dfrac{\lambda_{1S}}{\lambda_{1q}} \dfrac{\pd c_S}{\pd t} + \dfrac{\lambda_{1q_S}}{\lambda_{1q}} \left( \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{S2}}{\lambda_{Sq}} \dfrac{\pd c_2}{\pd t} + \dfrac{\lambda_{SS}}{\lambda_{Sq}} \dfrac{\pd c_S}{\pd t} \right) \right) \nonumber\\
                       %&= RI_1 + F \left( \dfrac{\lambda_{11}}{\lambda_{1q}} + \dfrac{\lambda_{1q_S}}{\lambda_{1q}} \dfrac{\lambda_{S1}}{\lambda_{Sq}} \right) \dfrac{\pd c_1}{\pd t} + F \left( \dfrac{\lambda_{12}}{\lambda_{1q}} + \dfrac{\lambda_{1q_S}}{\lambda_{1q}} \dfrac{\lambda_{S2}}{\lambda_{Sq}} \right) \dfrac{\pd c_2}{\pd t} + F \left( \dfrac{\lambda_{1S}}{\lambda_{1q}} + \dfrac{\lambda_{1q_S}}{\lambda_{1q}} \dfrac{\lambda_{SS}}{\lambda_{Sq}} \right) \dfrac{\pd c_S}{\pd t} \nonumber\\
                       %&= RI_1 - F \left( \left( \lambda_{11} + \dfrac{\lambda_{1q} \cdot \lambda_{S1}}{\lambda_{Sq}} \right) \dfrac{\pd c_1}{\pd t} \left( \lambda_{12} + \dfrac{\lambda_{1q} \cdot \lambda_{S2}}{\lambda_{Sq}} \right) \dfrac{\pd c_2}{\pd t} + \left( \lambda_{1S} + \dfrac{\lambda_{1q} \cdot \lambda_{SS}}{\lambda_{Sq}} \right) \dfrac{\pd c_S}{\pd t} \right) \nonumber\\
                       %&= \dfrac{RI_1}{1 + F \left( \lambda_{11} + \dfrac{\lambda_{1q} \cdot \lambda_{S1}}{\lambda_{Sq}} \right)} - \dfrac{F \left(\lambda_{12} + \dfrac{\lambda_{1q} \cdot \lambda_{S2}}{\lambda_{Sq}} \right)}{1 + F \left(\lambda_{11} + \dfrac{\lambda_{1q} \cdot \lambda_{S1}}{\lambda_{Sq}} \right)} \dfrac{\pd c_2}{\pd t} - \dfrac{F \left( \lambda_{1S} + \dfrac{\lambda_{1q} \cdot \lambda_{SS}}{\lambda_{Sq}} \right)}{1 + F \left( \lambda_{11} + \dfrac{\lambda_{1q} \cdot \lambda_{S1}}{\lambda_{Sq}} \right)} \dfrac{\pd c_S}{\pd t} \label{eq:dc_1}\\
\xrightarrow{(\ref{eq:MB_x}) + (\ref{eq:SMA-Mehr-dq2})} \nonumber\\ %dc1/dt
\dfrac{\pd c_2}{\pd t} &= RI_2 - F \dfrac{\pd q_2}{\pd t} \nonumber\\
                       &= RI_2 + F \left( \dfrac{\lambda_{21} \cdot \lambda_{Sq} - \lambda_{2q} \cdot \lambda_{S1}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_1}{\pd t} +  \dfrac{\lambda_{22} \cdot \lambda_{Sq} - \lambda_{2q} \cdot \lambda_{S2}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_2}{\pd t} + \dfrac{\lambda_{2S} \cdot \lambda_{Sq} - \lambda_{2q} \cdot \lambda_{SS}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_S}{\pd t} \right) \nonumber\\ %\label{eq:dc_2}
                       %&= RI_2 + F \left( \dfrac{\lambda_{21}}{\lambda_{2q}} \dfrac{\pd c_1}{\pd t} + \dfrac{\lambda_{22}}{\lambda_{2q}} \dfrac{\pd c_2}{\pd t} + \dfrac{\lambda_{2S}}{\lambda_{2q}} \dfrac{\pd c_S}{\pd t} + \dfrac{\lambda_{2q_S}}{\lambda_{2q}} \dfrac{\pd q_S}{\pd t} \right) \label{eq:dc_2}\\
                       %&= \frac{RI_2}{1 - F \dfrac{\lambda_{2q_S}}{\lambda_{2q}}} + \dfrac{F \dfrac{\lambda_{21}}{\lambda_{2q}}}{1 - F \dfrac{\lambda_{2q_S}}{\lambda_{2q}}} \dfrac{\pd c_1}{\pd t} + \dfrac{F \dfrac{\lambda_{22}}{\lambda_{2q}}}{1 - F \dfrac{\lambda_{2q_S}}{\lambda_{2q}}} \dfrac{\pd c_2}{\pd t} + \dfrac{F \dfrac{\lambda_{2S}}{\lambda_{2q}}}{1 - F \dfrac{\lambda_{2q_S}}{\lambda_{2q}}} \dfrac{\pd c_S}{\pd t} \nonumber\\
                       %&= \frac{RI_2 \cdot \lambda_{2q}}{\lambda_{2q} - F \cdot \lambda_{2q_S}} + \dfrac{F \cdot \lambda_{21}}{\lambda_{2q} - F \cdot \lambda_{2q_S}} \dfrac{\pd c_2}{\pd t} + \dfrac{F \cdot \lambda_{22}}{\lambda_{2q} - F \cdot \lambda_{2q_S}} \dfrac{\pd c_2}{\pd t} + \dfrac{F \cdot \lambda_{2S}}{\lambda_{2q} - F \cdot \lambda_{2q_S}} \dfrac{\pd c_S}{\pd t} \\
                       %&= RI_2 - F \left( \left( \lambda_{21} + \dfrac{\lambda_{2q} \cdot \lambda_{S1}}{\lambda_{Sq}} \right) \dfrac{\pd c_1}{\pd t} + \left( \lambda_{22} + \dfrac{\lambda_{2q} \cdot \lambda_{S2}}{\lambda_{Sq}} \right) \dfrac{\pd c_2}{\pd t} + \left(\lambda_{2S} + \dfrac{\lambda_{2q} \cdot \lambda_{SS}}{\lambda_{Sq}} \right) \dfrac{\pd c_S}{\pd t} \right) \nonumber\\
                       %&= \dfrac{RI_2}{1 + F \left( \lambda_{22} + \dfrac{\lambda_{2q} \cdot \lambda_{S2}}{\lambda_{Sq}} \right)} - \dfrac{F \left( \lambda_{21} + \dfrac{\lambda_{2q} \cdot \lambda_{S1}}{\lambda_{Sq}} \right)}{1 + F \left( \lambda_{22} + \dfrac{\lambda_{2q}\cdot\lambda_{S2}}{\lambda_{Sq}} \right)} \dfrac{\pd c_1}{\pd t} - \dfrac{F \left( \lambda_{2S} + \dfrac{\lambda_{2q} \cdot \lambda_{SS}}{\lambda_{Sq}} \right)}{1 + F \left( \lambda_{22} + \dfrac{\lambda_{2q} \cdot \lambda_{S2}}{\lambda_{Sq}} \right)} \dfrac{\pd c_S}{\pd t} \label{eq:dc_2}\\
\nonumber\\
\Longrightarrow
RI_S &= - F \dfrac{\lambda_{S1}}{\lambda_{Sq}} \dfrac{\pd c_1}{\pd t} - F \dfrac{\lambda_{S2}}{\lambda_{Sq}} \dfrac{\pd c_2}{\pd t} + \left( 1 - F \dfrac{\lambda_{SS}}{\lambda_{Sq}} \right) \dfrac{\pd c_S}{\pd t} \label{eq:dc_S}\\
%&= \left(1+F\dfrac{\lambda_{SS}}{\lambda_{Sq}}\right)\dfrac{\pd c_S}{\pd t}+F\dfrac{\lambda_{S1}}{\lambda_{Sq}}\dfrac{\pd c_1}{\pd t}+F\dfrac{\lambda_{S2}}{\lambda_{Sq}}\dfrac{\pd c_2}{\pd t}\\
RI_1 &= \left( 1 + F \dfrac{ \lambda_{1q} \cdot \lambda_{S1} - \lambda_{11} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \right) \dfrac{\pd c_1}{\pd t} + F \dfrac{\lambda_{1q} \cdot \lambda_{S2} - \lambda_{12} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_2}{\pd t} + F \dfrac{\lambda_{1q} \cdot \lambda_{SS} - \lambda_{1S} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_S}{\pd t} \label{eq:dc_1}\\
%&= F\left(\lambda_{1S}+\dfrac{\lambda_{1q}\cdot\lambda_{SS}}{\lambda_{Sq}}\right)\dfrac{\pd c_S}{\pd t}+\left(1+F\left(\lambda_{11}+\dfrac{\lambda_{1q}\cdot\lambda_{S1}}{\lambda_{Sq}}\right)\right)\dfrac{\pd c_1}{\pd t}+F\left(\lambda_{12}+\dfrac{\lambda_{1q}\cdot\lambda_{S2}}{\lambda_{Sq}}\right)\dfrac{\pd c_2}{\pd t}\\
RI_2 &= F \dfrac{\lambda_{2q} \cdot \lambda_{S1} - \lambda_{21} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_1}{\pd t} + \left( 1 + F \dfrac{\lambda_{2q} \cdot \lambda_{S2} - \lambda_{22} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \right) \dfrac{\pd c_2}{\pd t} + F \dfrac{\lambda_{2q} \cdot \lambda_{SS} - \lambda_{2S} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} \dfrac{\pd c_S}{\pd t} \label{eq:dc_2}\\
%&= F\left(\lambda_{2S}+\dfrac{\lambda_{2q}\cdot\lambda_{SS}}{\lambda_{Sq}}\right)\dfrac{\pd c_S}{\pd t}+F\left(\lambda_{21}+\dfrac{\lambda_{2q}\cdot\lambda_{S1}}{\lambda_{Sq}}\right)\dfrac{\pd c_1}{\pd t}+\left(1+F\left(\lambda_{22}+\dfrac{\lambda_{2q}\cdot\lambda_{S2}}{\lambda_{Sq}}\right)\right)\dfrac{\pd c_2}{\pd t}\\
\nonumber\\
\rightarrow
\overrightarrow{RI} &= \mathbf{G} \cdot \overrightarrow{\dfrac{\pd c}{\pd t}}\\
\textrm{mit } \mathbf{G} &= \left( \begin{array}{ccc}
1 - F \dfrac{\lambda_{SS}}{\lambda_{Sq}}& - F \dfrac{\lambda_{S1}}{\lambda_{Sq}} & - F \dfrac{\lambda_{S2}}{\lambda_{Sq}} \\
F \dfrac{\lambda_{1q} \cdot \lambda_{SS} - \lambda_{1S} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} & 1 + F \dfrac{\lambda_{1q} \cdot \lambda_{S1} - \lambda_{11} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} & F \dfrac{\lambda_{1q} \cdot \lambda_{S2} - \lambda_{12} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}}\\
F \dfrac{\lambda_{2q} \cdot \lambda_{SS} - \lambda_{2S} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} & F \dfrac{\lambda_{2q} \cdot \lambda_{S1} - \lambda_{21} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}} & 1 + F \dfrac{\lambda_{2q} \cdot \lambda_{S2} - \lambda_{22} \cdot \lambda_{Sq}}{\lambda_{Sq} \cdot \lambda_{q}}\\
% 1 + F \dfrac{\lambda_{SS}}{\lambda_{Sq}} & F\dfrac{\lambda_{S1}}{\lambda_{Sq}} & F \dfrac{ \lambda_{S2}}{\lambda_{Sq}}\\
% F \left( \lambda_{1S} + \dfrac{\lambda_{1q} \cdot \lambda_{SS}}{\lambda_{Sq}} \right) & 1 + F \left( \lambda_{11} + \dfrac{\lambda_{1q} \cdot \lambda_{S1}}{\lambda_{Sq}} \right) & F \left( \lambda_{12} + \dfrac{\lambda_{1q} \cdot \lambda_{S2}}{\lambda_{Sq}} \right)\\
% F\left(\lambda_{2S}+\dfrac{\lambda_{2q}\cdot\lambda_{SS}}{\lambda_{Sq}}\right) &1 + F \left( \lambda_{22} + \dfrac{\lambda_{2q} \cdot \lambda_{S2}}{\lambda_{Sq}} \right) &1 + F \left( \lambda_{22} + \dfrac{\lambda_{2q} \cdot \lambda_{S2}}{\lambda_{Sq}} \right)
\end{array}\right) \nonumber\\
\textrm{ und } \overrightarrow{\dfrac{\pd c}{\pd t}} &= \left(
\begin{array}{c}
\dfrac{\pd c_S}{\pd t}\\
\dfrac{\pd c_1}{\pd t}\\
\dfrac{\pd c_2}{\pd t}
\end{array}\right)
\textrm{ und } \overrightarrow{RI} = \left( \begin{array}{c}
RI_S\\
RI_1\\
RI_2
\end{array} \right)
\nonumber\\
\rightarrow \overrightarrow{\dfrac{\pd c}{\pd t}} &= \mathbf{G}^{ - 1} \cdot \overrightarrow{RI} \nonumber\\
\textrm{sei } \mathbf{G} &= \left( \begin{array}{ccc}
a&b&c\\
d&e&f\\
g&h&k
\end{array} \right) \nonumber\\
\rightarrow \mathbf{G}^{ - 1} &= \left( \begin{array}{ccc}
ek - fh & ch - bk & bf - ce\\
fg - dk & ak - cg & cd - af\\
dh - eg & bg - ah & ae - bd
\end{array} \right) \cdot \frac{1}{a \cdot (e \cdot k - f \cdot h) + b \cdot (f \cdot g - d \cdot k) + c \cdot (d \cdot h - e \cdot
g)}
\end{align}

%\subsection{Randbedingungen}
%\begin{align}
%\mathbf{M} \cdot \overrightarrow{c} &= \overrightarrow{h} \nonumber\\
%\textrm{wobei } \mathbf{M} &= \left( \begin{array}{ccccccccc}
%D_L \cdot A_{0,0} - u \cdot L \cdot Le & D_L \cdot A_{0,1} & D_L \cdot A_{0,2} & 0         & 0                 & 0         & 0                 & 0         & 0 \\
%A_{2,0}                                & A_{2,1}           & A_{2,2} - A_{0,0} & - A_{0,1} & - A_{0,2}         & 0         & 0                 & 0         & 0 \\
%0                                      & 0                 & A_{2,0}           & A_{2,1}   & A_{2,2} - A_{0,0} & - A_{0,1} & - A_{0,2}         & 0         & 0 \\
%0                                      & 0                 & 0                 & 0         & A_{2,0}           & A_{2,1}   & A_{2,2} - A_{0,0} & - A_{0,1} & - A_{0,2}\\
%0                                      & 0                 & 0 & 0
%&0                  & 0         & A_{2,0}           & A_{2,1}   &
%A_{2,2} \end{array} \right)
%\nonumber\\
%\textrm{und} \overrightarrow{c} &= \left( \begin{array}{c}
%c^0_0\\
%c^0_1\\
%c^1_0\\
%c^1_1\\
%c^2_0\\
%c^2_1\\
%c^3_0\\
%c^3_1\\
%c^3_2 \end{array} \right), \overrightarrow{h} = \left( \begin{array}{c} - u \cdot L \cdot Le \cdot c_f\\
%0\\
%0\\
%0\\
%0\\
%0\\
%0\\
%0\\
%0 \end{array} \right) \textrm{alle c sind unbekannte} \nonumber
%\end{align}

\end{document}
